Arakelov几何（Arakelov Geometry）（影印版）

样章

作者：

Atsushi Moriwaki

定价：

135.00 元

版面字数：

500.00千字

开本：

16开

装帧形式：

精装

页数：

304

最新
印次时间：

2026年03月

ISBN：

978-7-04-065929-0

物料号：

65929-00

出版时间：

2026-03-27

读者对象：

学术著作

一级分类：

自然科学

二级分类：

数学与统计

三级分类：

代数几何学

暂无

目录
- 前辅文
- Preface
- Chapter 1. Preliminaries
  - 1.1. Frequently used notation and conventions
  - 1.2. Normed finite-dimensional vector space
  - 1.3. Lemmas on the length of modules
  - 1.4. Image of a homomorphism and its determinant
  - 1.5. Norm of flat and finite homomorphisms
  - 1.6. Principal divisor and Weil’s reciprocity law
  - 1.7. Existence of rational sections not passing through given points
  - 1.8. Graded modules and ample invertible sheaves
  - 1.9. Several results on separable extensions of the base field
  - 1.10. Determinant bundle
  - 1.11. Complex manifold and Hodge theory
  - 1.12. Connection and curvature
  - 1.13. Poincaré-Lelong formula
  - 1.14. C∞ on reduced complex space
- Chapter 2. Geometry of Numbers
  - 2.1. Convex set and Minkowski’s theorem
  - 2.2. Polar dual set and Mahler’s inequality
  - 2.3. The Brunn-Minkowski theorem
  - 2.4. Estimate of the number of points in a convex lattice
  - 2.5. Normed finitely generated Z-module
  - 2.6. λQ and λZ
- Chapter 3. Arakelov Geometry on Arithmetic Curves
  - 3.1. Orders
  - 3.2. Arithmetic Chow group over a reduced order
  - 3.3. Hermitian R-module
  - 3.4. Arithmetic Riemann-Roch formula on arithmetic curves
  - 3.5. Effective estimate of the number of small sections
  - 3.6. Several formulae on arithmetic degree
  - 3.7. Volume exactness
  - 3.8. Ample invertible sheaves on arithmetic curves
- Chapter 4. Arakelov Geometry on Arithmetic Surfaces
  - 4.1. Deligne’s pairing
  - 4.2. Green functions on Riemann surfaces
  - 4.3. Arithmetic Chow groups on arithmetic surfaces
  - 4.4. Intersection theory on arithmetic surfaces
  - 4.5. Arakelov metric of dualizing sheaf and adjunction formula
  - 4.6. Determinant bundles for curves
  - 4.7. Faltings’ Riemann-Roch theorem on arithmetic surfaces
  - 4.8. Determinant bundle and theta divisor
  - 4.9. Existence of Faltings’ metric
- Chapter 5. Arakelov Geometry on General Arithmetic Varieties
  - 5.1. Preliminaries on algebraic geometry and complex geometry
  - 5.2. Intersection theory of Cartier divisors on excellent schemes
  - 5.3. Higher dimensional generalization of Weil’s reciprocity law in complex geometry
  - 5.4. Intersection theory on arithmetic varieties
  - 5.5. Characteristic classes and Bott-Chern secondary characteristic form
  - 5.6. Arithmetic characteristic classes
  - 5.7. Arithmetic Riemann-Roch formula
  - 5.8. Multi-indexed version of Gromov’s inequality
  - 5.9. Arithmetic Hilbert-Samuel formula
  - 5.10. Several kinds of positivity of C∞-hermitian invertible sheaves
  - 5.11. Estimation of λQ for a normed graded ring
- Chapter 6. Arithmetic Volume Function and Its Continuity
  - 6.1. Arithmetic volume function
  - 6.2. Extension of volume function over Q
  - 6.3. Continuity of volume function
  - 6.4. Generalized Hodge index theorem
  - 6.5. Estimate of the number of small sections
- Chapter 7. Nakai-Moishezon Criterion on an Arithmetic Variety
  - 7.1. Endmorphism N and its basic properties
  - 7.2. Bounded extension of holomorphic sections
  - 7.3. Proof of Nakai-Moishezon’s criterion on an arithmetic variety
  - 7.4. Arithmetic Hilbert-Samuel formula
- Chapter 8. Arithmetic Bogomolov Inequality
  - 8.1. Semistable locally free coherent sheaves on algebraic curves
  - 8.2. Hermite-Einstein metric and stability
  - 8.3. Arithmetic Bogomolov inequality and its proof
- Chapter 9. Lang-Bogomolov Conjecture
  - 9.1. Height function
  - 9.2. Height function on abelian variety
  - 9.3. Equidistribution theorem
  - 9.4. Cubic metric on complex abelian variety
  - 9.5. Bogomolov’s conjecture
  - 9.6. The Lang-Bogomolov Conjecture
  - 9.7. Small points with respect to a subgroup of finite rank
  - 9.8. The proof of Theorem 9.24
- Bibliography
- Index